A relationship between the Dirichlet and Regularity Problems for elliptic equations

Abstract

We study the relationship between the solvability of the $L^p$ Dirichlet problem $(D{)}_p$ and that of the $L^q$ regularity problem $(R{)}_q$ for second order elliptic equations with bounded measurable coefficients. It is known that the solvability of $(R{)}_p$ implies that of $(D{)}_{p^{\prime }}$. In this note we show that if $(D{)}_{p^{\prime }}$ is solvable, then either $(R{)}_p$ is solvable or $(R{)}_q$ is not solvable for any $1<q<\infty$.